CMSC 351 - Thought Questions #1
Solve each of the following to a specific number without the use
of a calculator:
(a) log2256
(b) Given logab = 23 and logad = 15
compute loga(b/d)
(c) log7(494)
(d) log2(1/8)
Find the integrals of each of the following:
(hint: think integration-by-parts and log rules for some of them)
(note: loge is also known as ln)
(e) x4
(f) logex3
(g) x logex
(h) Prove the following by mathematical induction:
n
Σ (2i + 5/i) < (2n2+15)
i=1
Show your Base Case(s):
State your Inductive Hypothesis:
State what you plan to prove in your Inductive Step:
Prove it:
(i) Prove the following by mathematical induction:
n n(n+1)(n+2)
Σ i(i+1) = ------------
i=1 3
Show your Base Case(s):
State your Inductive Hypothesis:
State what you plan to prove in your Inductive Step:
Prove it:
(j) Determine whether the following is true or false, and then prove that.
3 | (n3+2n)
(k) Consider the following:
Assuming we are looking at comparisons, and working on the
assumption that you can perform 220 comparisons per second,
consider approximately how large n could be so that you
could solve a problem of each of the following complexities
in the specified time period.
|
1 Second |
1 Minute |
1 Hour |
1 Day |
1 Month |
1 Year |
1 Century |
log2n | | | | | | | |
n | | | | | | | |
n log2n | | | | | | | |
n2 | | | | | | | |
n3 | | | | | | | |
2n | | | | | | | |
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